P2.T5.200. Implied volatility

200.1. When the current price of a non-dividend stock is $30.00, an in-the-money (ITM) European call option with one (1.0) year to expiration and a strike price of $25.00 exhibits an implied volatility of 42.0%, although the historical volatility of the stock is 36.0%. Which is nearest to the implied volatility of an out-of-the money (OTM) put option on the same stock, assuming the same one year maturity and current stock price?

a. 36%
b. 42%
c. 48%
d. Unclear without traded price of put

200.2. Assume that implied volatility, on the y-axis, is plotted against the ratio of strike price to current stock price, K/S(0), on the x-axis. Options on a certain equity index exhibit a typical implied volatility skew: 44% at K/S = 0.7, 39% at K/S (ATM) = 1.0, and 28% at K/S = 1.3. Each of the following statements is consistent with this observed implied volatility skew EXCEPT for:

a. Risk-neutral log returns are heavy-tailed on the lower tail but light-tailed on the upper tail
b. Investors are highly risk-averse and willing to pay more for extreme tail insurance
c. Traders are optimistic and want to write more deeply out-of-the-money (OTM) puts; i.e., increase in supply of OTM puts
d. Traders are pessimistic and generally assign lower probabilities to the payout of deeply out-of-the-money (OTM) call options

200.3. A European put option with a strike price of (K) and maturity of six months has an implied volatility of 22%. Simultaneously, a European call option on the same stock, with an identical strike price of (K) and identical maturity of six months, has an implied volatility of 29%. What is the arbitrage trade?

a. Buy the call, sell the put, short the stock, and lend cash at the riskfree rate
b. Buy the put, sell the call, buy the stock, and borrow cash at the riskfree rate
c. There is not necessarily an arbitrage trade; it depends on respective prices of the call and put (which are not given)
d. There is not necessarily an arbitrage trade; it depends on the validity of the lognormal stock price assumption